Their dismissal left just four of us to cover two hospitals and the work of six junior residents. I quickly did the math in my head. We would have to split up into pairs, and we would have to work through every other night for the rest of the year.

The prospect of such a grueling timetable inspired us to come up with equally extreme solutions. One month we tried “power 60s,” working a 60-hour shift every other weekend so that the other of the pair could have at least one full day’s break.

That experiment didn’t last long.

But my most enduring memory of that year is not the exhaustion; it is the panic and anxiety that enveloped us as we struggled to cover far more work than four people — however willing — could reasonably complete. Our lives had been distilled down to a simple math equation with three variables: the work to be done, the time needed to complete it and the number of people available to do it. We worked longer hours not because our senior surgeons told us to, but because that was the only way we could balance the equation.

If only doctors in training these days had it so easy.

Over the past decade, in response to public concerns about medical errors arising from fatigue, the Accreditation Council for Graduate Medical Education, the organization responsible for accrediting American medical residency programs, has been progressively limiting the number of hours that trainees can work. The latest mandate, which took effect in 2011, is the most stringent and deals most specifically with interns. These youngest doctors are allowed to work no longer than 16 hours in a day; and residency programs that violate the restriction risk losing their accreditation.

In response to the 16-hour mandate and faced with a Rubik’s-cube conundrum of covering all the work with the same number of interns working fewer hours, training programs across the country came up with several innovative scheduling configurations. Some created complicated and overlapping shifts where outgoing doctors “signed out” their patients, passing off their responsibilities to the incoming shift. Others adopted a “night float” system that meant a resident just a year out of internship had to carry the work of as many as 12 interns at night, looking after more than 100 patients and fielding questions about those patients at best every 20 minutes and at worst every 11 minutes throughout the night.

Now, two years after the 16-hour mandate was established, studies on the outcomes are being published, and the results reveal one thing.

Maybe we should have thought a little harder about the arithmetic.

Contrary to expectations, these studies have shown that interns have not been getting significantly more sleep. Moreover, they are not happier, nor are they studying more. In one national survey, nearly half of all doctors in training disapproved of the regulations altogether. Another study revealed that interns were spending less time in educational activities because the additional time required for such conferences and lectures would push them over the 16-hour limit.

In addition, there has been no significant improvement in the quality of care since the work limits took effect. In one case, doctors had to scrap the night float system because the nurses thought the care offered by trainees on that schedule was so poor. Another study revealed that interns confessed to having more concerns about making serious medical errors after the mandate than before.

Health care trends over the last two decades have only exacerbated young doctors’ workload. Admissions to teaching hospitals increased nearly 50 percent from 1990 to 2010; in that same period, the number of doctors in training available to do the work increased by only 10 percent. And because insurers are pushing for shorter hospital stays, only the sickest patients, many of whom require complex care, remain hospitalized.

“It’s as if you told airline pilots that they could only work a certain number of hours, but they had to fly 50 percent more flights,” Dr. Goitein said.

The accreditation council plans to review the findings from these studies, but one important message is clear: More funding needs to be devoted to increasing the number of doctors in training and finding ways to lighten their workload. While these efforts may cost more initially, Dr. Goitein notes that several small studies already indicate that giving these doctors more reasonable workloads can result in significant improvements in quality. “It could be a short-term investment for long-term gains.”

“You can’t keep asking these young doctors to do more and more work in less time without affecting patient care,” Dr. Goitein said. “Until we address the problem of overwork, we’re just playing a shell game.”

A version of this article appears in print on 04/23/2013, on page D5 of the NewYork edition with the headline: Math Doesn’t Add Up on Trainees’ Hours.